Civil Engineering Reference
In-Depth Information
Table 2.1 Semi-discretisation of partial differential
equations
Term in
Typical term in
Symmetry?
differential equation
matrix equation
N
i
N
j
d
x
u
Ye s
N
i
d
N
j
d
x
d
u
d
x
d
x
No
−
d
N
j
d
x
d
2
u
d
N
i
d
x
d
x
Ye s
2
d
x
d
2
d
2
d
4
N
j
d
x
N
i
d
x
u
2
d
x
Ye s
4
2
d
x
2.8 Alternative derivation of element stiffness
Instead of working from the governing differential equation, element properties can often
be derived by an alternative method based on a consideration of energy. For example, the
strain energy stored due to bending of a very small length
δx
of the elastic beam element
in Figure 2.2 is,
2
EI
δx
1
2
M
δU
=
(2.38)
where
M
is the “bending moment” and by conservation of energy this must be equal to
the work done by the external loads
q
, thus
1
2
qwδx
δW
=
(2.39)
The bending moment
M
is related to
w
through the “moment-curvature” expression,
EI
d
2
w
d
x
M
=−
2
or
M
=
[
D
]
{
A
}
w
(2.40)
d
2
2
. Writing (2.38) in
where [
D
] is the material property
EI
and
{
A
}
is the operator
−
/
d
x
the form,
d
2
1
2
w
d
x
δU
=
−
Mδx
(2.41)
2
we have
1
2
(
{
T
δU
=
A
}
w)
Mδx
(2.42)
